The document proposes novel methods for generating self-invertible matrices for use in the Hill cipher encryption algorithm. It discusses how the Hill cipher works and the issue that the encryption matrix may not be invertible, preventing decryption. It then presents three methods for generating self-invertible matrices of sizes 2x2, 3x3, and 4x4 that can be used as the encryption key matrix in Hill cipher to allow for decryption without needing to calculate the inverse matrix.
EFFICIENT DIGITAL ENCRYPTION ALGORITHM BASED ON MATRIX SCRAMBLING TECHNIQUEIJNSA Journal
This paper puts forward a safe mechanism of data transmission to tackle the security problem of information which is transmitted in Internet. We propose a new technique on matrix scrambling which is based on random function, shifting and reversing techniques of circular queue. We give statistical analysis, sequence random analysis, and sensitivity analysis to plaintext and key on the proposed scheme. The experimental results show that the new scheme has a very fast encryption speed and the key space is expanded and it can resist all kinds of cryptanalytic, statistical attacks, and especially, our new method can be also used to solve the problem that is easily exposed to chosen plaintext attack. We give our detailed report to this algorithm, and reveal the characteristic of this algorithm by utilizing an example.
Information Content of Complex NetworksHector Zenil
This short talk given in Stockholm, Sweden, explains how algorithmic complexity measures, notably Kolmogorov complexity approximated both by lossless compression algorithms and the Block Decomposition Method (BDM) are capable of characterizing graphs and networks by some of their group-theoretic and topological properties, notably graph automorphism group size and clustering coefficients of complex networks. The method distinguished between models of networks such as regular, random, small-world and scale-free.
Low Power 32×32 bit Multiplier Architecture based on Vedic Mathematics Using ...VIT-AP University
In this paper the most significant aspect of the proposed method is that, the developed multiplier architecture is based on vertical and crosswise structure of Ancient Indian Vedic Mathematics. As per this proposed architecture, for two 32-bit numbers; the multiplier and multiplicand, each are grouped as 16-bit numbers so that it decomposes into 16×16 multiplication modules. It is also illustrated that the further hierarchical decomposition of 8×8 modules into 4×4 modules and then 2×2 modules will have a significant VHDL coding of for 32x32 bits multiplication and their used FPGA family Virtex 7 low power implementation by Xilinx Synthesis 16.1 tool done. The synthesis results show that the computation time for calculating the product of 32x32 bits is delay 29.256 ns. (11.499ns logic, 11.994ns route) (48.9% logic, 51.1% route).
Slides from my session on GPars at Gr8Conf EU 2012. It was a manifesto speech for using actors, dataflow, CSP and data parallelism and avoiding all explicit locking and indeed explicit shared-memory multi-threading.
An evolutionary method for constructing complex SVM kernelsinfopapers
D. Simian, F. Stoica, An Evolutionary Method for Constructing Complex SVM Kernels, Recent Advances in Mathematics and Computers in Biology and Chemistry, Proceedings of the 10th International Conference on Mathematics and Computers in Biology and Chemistry, MCBC’09, Prague, Chech Republic, WSEAS Press, ISBN 978-960-474-062-8, ISSN 1790-5125, pp.172-178, 2009
EFFICIENT DIGITAL ENCRYPTION ALGORITHM BASED ON MATRIX SCRAMBLING TECHNIQUEIJNSA Journal
This paper puts forward a safe mechanism of data transmission to tackle the security problem of information which is transmitted in Internet. We propose a new technique on matrix scrambling which is based on random function, shifting and reversing techniques of circular queue. We give statistical analysis, sequence random analysis, and sensitivity analysis to plaintext and key on the proposed scheme. The experimental results show that the new scheme has a very fast encryption speed and the key space is expanded and it can resist all kinds of cryptanalytic, statistical attacks, and especially, our new method can be also used to solve the problem that is easily exposed to chosen plaintext attack. We give our detailed report to this algorithm, and reveal the characteristic of this algorithm by utilizing an example.
Information Content of Complex NetworksHector Zenil
This short talk given in Stockholm, Sweden, explains how algorithmic complexity measures, notably Kolmogorov complexity approximated both by lossless compression algorithms and the Block Decomposition Method (BDM) are capable of characterizing graphs and networks by some of their group-theoretic and topological properties, notably graph automorphism group size and clustering coefficients of complex networks. The method distinguished between models of networks such as regular, random, small-world and scale-free.
Low Power 32×32 bit Multiplier Architecture based on Vedic Mathematics Using ...VIT-AP University
In this paper the most significant aspect of the proposed method is that, the developed multiplier architecture is based on vertical and crosswise structure of Ancient Indian Vedic Mathematics. As per this proposed architecture, for two 32-bit numbers; the multiplier and multiplicand, each are grouped as 16-bit numbers so that it decomposes into 16×16 multiplication modules. It is also illustrated that the further hierarchical decomposition of 8×8 modules into 4×4 modules and then 2×2 modules will have a significant VHDL coding of for 32x32 bits multiplication and their used FPGA family Virtex 7 low power implementation by Xilinx Synthesis 16.1 tool done. The synthesis results show that the computation time for calculating the product of 32x32 bits is delay 29.256 ns. (11.499ns logic, 11.994ns route) (48.9% logic, 51.1% route).
Slides from my session on GPars at Gr8Conf EU 2012. It was a manifesto speech for using actors, dataflow, CSP and data parallelism and avoiding all explicit locking and indeed explicit shared-memory multi-threading.
An evolutionary method for constructing complex SVM kernelsinfopapers
D. Simian, F. Stoica, An Evolutionary Method for Constructing Complex SVM Kernels, Recent Advances in Mathematics and Computers in Biology and Chemistry, Proceedings of the 10th International Conference on Mathematics and Computers in Biology and Chemistry, MCBC’09, Prague, Chech Republic, WSEAS Press, ISBN 978-960-474-062-8, ISSN 1790-5125, pp.172-178, 2009
LATTICE BASED TOOLS IN CRYPTANALYSIS FOR PUBLIC KEY CRYPTOGRAPHY IJNSA Journal
Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Lattice reduction has been successfully utilizing in Number Theory, Linear algebra and Cryptology. Not only the existence of lattice based cryptosystems of hard in nature, but also has vulnerabilities by lattice reduction techniques. In this survey paper, we are focusing on point lattices and then describing an introduction to
the theoretical and practical aspects of lattice reduction. Finally, we describe the applications of lattice reduction in Number theory, Linear algebra.
A Mixed Binary-Real NSGA II Algorithm Ensuring Both Accuracy and Interpretabi...IJECEIAES
In this work, a Neuro-Fuzzy Controller network, called NFC that implements a Mamdani fuzzy inference system is proposed. This network includes neurons able to perform fundamental fuzzy operations. Connections between neurons are weighted through binary and real weights. Then a mixed binaryreal Non dominated Sorting Genetic Algorithm II (NSGA II) is used to perform both accuracy and interpretability of the NFC by minimizing two objective functions; one objective relates to the number of rules, for compactness, while the second is the mean square error, for accuracy. In order to preserve interpretability of fuzzy rules during the optimization process, some constraints are imposed. The approach is tested on two control examples: a single input single output (SISO) system and a multivariable (MIMO) system.
A Numerical Method for the Evaluation of Kolmogorov Complexity, An alternativ...Hector Zenil
We present a novel alternative method (other than using compression algorithms) to approximate the algorithmic complexity of a string by calculating its algorithmic probability and applying Chaitin-Levin's coding theorem.
Algorithmic Information Theory and Computational BiologyHector Zenil
I present cutting-edge concepts and tools drawn from algorithmic information theory (AIT) for new generation genetic sequencing, network biology and bioinformatics in general. AIT is the most advanced mathematical theory of information theory formally characterising the concepts and differences between simplicity, randomness and structure. Measures of AIT will empower computational medicine and systems biology to deal with big data, sophisticated analytics and a powerful new understanding framework.
In the classical model, the fundamental building block is represented by bits exists in two states a 0 or a 1. Computations are done by logic gates on the bits to produce other bits. By increasing the number of bits, the complexity of problem and the time of computation increases. A quantum algorithm is a sequence of operations on a register to transform it into a state which when measured yields the desired result. This paper provides introduction to quantum computation by developing qubit, quantum gate and quantum circuits.
Towards a stable definition of Algorithmic RandomnessHector Zenil
Although information content is invariant up to an additive constant, the range of possible additive constants applicable to programming languages is so large that in practice it plays a major role in the actual evaluation of K(s), the Kolmogorov complexity of a string s. We present a summary of the approach we've developed to overcome the problem by calculating its algorithmic probability and evaluating the algorithmic complexity via the coding theorem, thereby providing a stable framework for Kolmogorov complexity even for short strings. We also show that reasonable formalisms produce reasonable complexity classifications.
GENERAL REGRESSION NEURAL NETWORK BASED POS TAGGING FOR NEPALI TEXTcscpconf
This article presents Part of Speech tagging for Nepali text using General Regression Neural
Network (GRNN). The corpus is divided into two parts viz. training and testing. The network is
trained and validated on both training and testing data. It is observed that 96.13% words are
correctly being tagged on training set whereas 74.38% words are tagged correctly on testing
data set using GRNN. The result is compared with the traditional Viterbi algorithm based on
Hidden Markov Model. Viterbi algorithm yields 97.2% and 40% classification accuracies on
training and testing data sets respectively. GRNN based POS Tagger is more consistent than the
traditional Viterbi decoding technique.
Improving Performance of Back propagation Learning Algorithmijsrd.com
The standard back-propagation algorithm is one of the most widely used algorithm for training feed-forward neural networks. One major drawback of this algorithm is it might fall into local minima and slow convergence rate. Natural gradient descent is principal method for solving nonlinear function is presented and is combined with the modified back-propagation algorithm yielding a new fast training multilayer algorithm. This paper describes new approach to natural gradient learning in which the number of parameters necessary is much smaller than the natural gradient algorithm. This new method exploits the algebraic structure of the parameter space to reduce the space and time complexity of algorithm and improve its performance.
One of the fundamental issues in computer science is ordering a list of items. Although there is a number of sorting algorithms, sorting problem has attracted a great deal of research, because efficient sorting is important to optimize the use of other algorithms. This paper presents a new sorting algorithm which runs faster by decreasing the number of comparisons by taking some extra memory. In this algorithm we are using lists to sort the elements. This algorithm was analyzed, implemented and tested and the results are promising for a random data
Symmetric Key Generation Algorithm in Linear Block Cipher Over LU Decompositi...ijtsrd
In symmetric key algorithm in linear block cipher to encrypt and decrypt the messages using matrix and inverse matrix. In this proposed technique generate lower and upper triangular matrices from the square matrix using decomposition. In encryption process, the key is a lower triangular matrix and decryption process, the key is upper triangular matrix under modulation of the prime number. We illustrate the proposed technique with help of examples. P.Sundarayya | M.G.Vara Prasad"Symmetric Key Generation Algorithm in Linear Block Cipher Over LU Decomposition Method " Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-4 , June 2017, URL: http://www.ijtsrd.com/papers/ijtsrd90.pdf http://www.ijtsrd.com/computer-science/computer-security/90/symmetric-key-generation-algorithm-in--linear-block-cipher-over-lu-decomposition-method--/psundarayya-
LATTICE BASED TOOLS IN CRYPTANALYSIS FOR PUBLIC KEY CRYPTOGRAPHY IJNSA Journal
Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Lattice reduction has been successfully utilizing in Number Theory, Linear algebra and Cryptology. Not only the existence of lattice based cryptosystems of hard in nature, but also has vulnerabilities by lattice reduction techniques. In this survey paper, we are focusing on point lattices and then describing an introduction to
the theoretical and practical aspects of lattice reduction. Finally, we describe the applications of lattice reduction in Number theory, Linear algebra.
A Mixed Binary-Real NSGA II Algorithm Ensuring Both Accuracy and Interpretabi...IJECEIAES
In this work, a Neuro-Fuzzy Controller network, called NFC that implements a Mamdani fuzzy inference system is proposed. This network includes neurons able to perform fundamental fuzzy operations. Connections between neurons are weighted through binary and real weights. Then a mixed binaryreal Non dominated Sorting Genetic Algorithm II (NSGA II) is used to perform both accuracy and interpretability of the NFC by minimizing two objective functions; one objective relates to the number of rules, for compactness, while the second is the mean square error, for accuracy. In order to preserve interpretability of fuzzy rules during the optimization process, some constraints are imposed. The approach is tested on two control examples: a single input single output (SISO) system and a multivariable (MIMO) system.
A Numerical Method for the Evaluation of Kolmogorov Complexity, An alternativ...Hector Zenil
We present a novel alternative method (other than using compression algorithms) to approximate the algorithmic complexity of a string by calculating its algorithmic probability and applying Chaitin-Levin's coding theorem.
Algorithmic Information Theory and Computational BiologyHector Zenil
I present cutting-edge concepts and tools drawn from algorithmic information theory (AIT) for new generation genetic sequencing, network biology and bioinformatics in general. AIT is the most advanced mathematical theory of information theory formally characterising the concepts and differences between simplicity, randomness and structure. Measures of AIT will empower computational medicine and systems biology to deal with big data, sophisticated analytics and a powerful new understanding framework.
In the classical model, the fundamental building block is represented by bits exists in two states a 0 or a 1. Computations are done by logic gates on the bits to produce other bits. By increasing the number of bits, the complexity of problem and the time of computation increases. A quantum algorithm is a sequence of operations on a register to transform it into a state which when measured yields the desired result. This paper provides introduction to quantum computation by developing qubit, quantum gate and quantum circuits.
Towards a stable definition of Algorithmic RandomnessHector Zenil
Although information content is invariant up to an additive constant, the range of possible additive constants applicable to programming languages is so large that in practice it plays a major role in the actual evaluation of K(s), the Kolmogorov complexity of a string s. We present a summary of the approach we've developed to overcome the problem by calculating its algorithmic probability and evaluating the algorithmic complexity via the coding theorem, thereby providing a stable framework for Kolmogorov complexity even for short strings. We also show that reasonable formalisms produce reasonable complexity classifications.
GENERAL REGRESSION NEURAL NETWORK BASED POS TAGGING FOR NEPALI TEXTcscpconf
This article presents Part of Speech tagging for Nepali text using General Regression Neural
Network (GRNN). The corpus is divided into two parts viz. training and testing. The network is
trained and validated on both training and testing data. It is observed that 96.13% words are
correctly being tagged on training set whereas 74.38% words are tagged correctly on testing
data set using GRNN. The result is compared with the traditional Viterbi algorithm based on
Hidden Markov Model. Viterbi algorithm yields 97.2% and 40% classification accuracies on
training and testing data sets respectively. GRNN based POS Tagger is more consistent than the
traditional Viterbi decoding technique.
Improving Performance of Back propagation Learning Algorithmijsrd.com
The standard back-propagation algorithm is one of the most widely used algorithm for training feed-forward neural networks. One major drawback of this algorithm is it might fall into local minima and slow convergence rate. Natural gradient descent is principal method for solving nonlinear function is presented and is combined with the modified back-propagation algorithm yielding a new fast training multilayer algorithm. This paper describes new approach to natural gradient learning in which the number of parameters necessary is much smaller than the natural gradient algorithm. This new method exploits the algebraic structure of the parameter space to reduce the space and time complexity of algorithm and improve its performance.
One of the fundamental issues in computer science is ordering a list of items. Although there is a number of sorting algorithms, sorting problem has attracted a great deal of research, because efficient sorting is important to optimize the use of other algorithms. This paper presents a new sorting algorithm which runs faster by decreasing the number of comparisons by taking some extra memory. In this algorithm we are using lists to sort the elements. This algorithm was analyzed, implemented and tested and the results are promising for a random data
Symmetric Key Generation Algorithm in Linear Block Cipher Over LU Decompositi...ijtsrd
In symmetric key algorithm in linear block cipher to encrypt and decrypt the messages using matrix and inverse matrix. In this proposed technique generate lower and upper triangular matrices from the square matrix using decomposition. In encryption process, the key is a lower triangular matrix and decryption process, the key is upper triangular matrix under modulation of the prime number. We illustrate the proposed technique with help of examples. P.Sundarayya | M.G.Vara Prasad"Symmetric Key Generation Algorithm in Linear Block Cipher Over LU Decomposition Method " Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-4 , June 2017, URL: http://www.ijtsrd.com/papers/ijtsrd90.pdf http://www.ijtsrd.com/computer-science/computer-security/90/symmetric-key-generation-algorithm-in--linear-block-cipher-over-lu-decomposition-method--/psundarayya-
WEAKNESS ON CRYPTOGRAPHIC SCHEMES BASED ON REGULAR LDPC CODESIJNSA Journal
We propose a method to recover the structure of a randomly permuted chained code and how to cryptanalyse cryptographic schemes based on these kinds of error coding. As application of these methods is a cryptographic schema using regular Low Density Parity Check (LDPC) Codes. This result prohibits the use of chained code and particularly regular LDPC codes on cryptography
On the Usage of Chained Codes in CryptographyCSCJournals
We discuss the chained randomized linear code and their use in cryptography. We show that the adoption of randomized chained codes in the framework of McEliece cryptosystem expose the cryptosystem to some new attacks.
Image Encryption Using Advanced Hill Cipher AlgorithmIDES Editor
The Hill cipher algorithm is one of the symmetric
key algorithms that have several advantages in data
encryption. But, the inverse of the key matrix used for
encrypting the plaintext does not always exist. Then if the
key matrix is not invertible, then encrypted text cannot be
decrypted. In the Involutory matrix generation method the
key matrix used for the encryption is itself invertible. So, at
the time of decryption we need not to find the inverse of the
key matrix. The objective of this paper is to encrypt an
image using a technique different from the conventional Hill
Cipher. In this paper a novel advanced Hill (AdvHill)
encryption technique has been proposed which uses an
involutory key matrix. The scheme is a fast encryption
scheme which overcomes problems of encrypting the images
with homogeneous background. A comparative study of the
proposed encryption scheme and the existing scheme is
made. The output encrypted images reveal that the
proposed technique is quite reliable and robust.
Elliptic curve cryptography is additional powerful than different methodology that gains countless attention within the industry and plays vital role within the world of CRYPTOGRAPHY. This paper explains the strategy of elliptic curve cryptography victimization matrix scrambling method. during this methodology of cryptography we have a tendency to initial rework the plain text to elliptic curve so victimization matrix scrambling methodology we have a tendency to encrypt/decrypt the message. This method keeps information safe from unwanted attack to our information.
MESSAGE EMBEDDED CIPHER USING 2-D CHAOTIC MAPijccmsjournal
This paper constructs two encryption methods using 2-D chaotic maps, Duffings and Arnold’s cat maps
respectively. Both of the methods are designed using message embedded scheme and are analyzed for their validity, for plaintext sensitivity, key sensitivity, known plaintext and brute-force attacks. Due to the
less key space generally many chaotic cryptosystem developed are found to be weak against Brute force attack which is an essential issue to be solved. For this issue, concept of identifiability proved to be a necessary condition to be fulfilled by the designed chaotic cipher to resist brute force attack, which is a basic attack. As 2-D chaotic maps provide more key space than 1-D maps thus they are considered to be more suitable. This work is accompanied with analysis results obtained from these developed cipher. Moreover, identifiable keys are searched for different input texts at various key values.
The methods are found to have good key sensitivity and possess identifiable keys thus concluding that they can resist linear attacks and brute-force attacks.
Message Embedded Cipher Using 2-D Chaotic Mapijccmsjournal
This paper constructs two encryption methods using 2-D chaotic maps, Duffings and Arnold’s cat maps
respectively. Both of the methods are designed using message embedded scheme and are analyzed for
their validity, for plaintext sensitivity, key sensitivity, known plaintext and brute-force attacks. Due to the
less key space generally many chaotic cryptosystem developed are found to be weak against Brute force
attack which is an essential issue to be solved. For this issue, concept of identifiability proved to be a
necessary condition to be fulfilled by the designed chaotic cipher to resist brute force attack, which is a
basic attack. As 2-D chaotic maps provide more key space than 1-D maps thus they are considered to be
more suitable. This work is accompanied with analysis results obtained from these developed cipher.
Moreover, identifiable keys are searched for different input texts at various key values.
the use of digital data has been increase over the past decade which has led to the evolution of digital world. With this evolution the use of data such as text, images and other multimedia for communication purpose over network needs to be secured during transmission. Images been the most extensively used digital data throughout the world, there is a need for the security of images, so that the confidentiality, integrity and availability of the data is maintained. There is various cryptography techniques used for image security of which the asymmetric cryptography is most extensively used for securing data transmission. This paper discusses about Elliptic Curve Cryptography an asymmetric public key cryptography method for image transmission. With security it is also crucial to address the computational aspects of the cryptography methods used for securing images. The paper proposes an Image encryption and decryption method using ECC. Integrity of image transmission is achieved by using Elliptic Curve Digital Signature Algorithm (ECDSA) and also considering computational aspects at each stage.
Improved Caesar Cipher with Random Number Generation Technique and Multistage...ijcisjournal
Secured Communication involves Encryption process at the sending end and Decryption process at the receiving end of the communication system. Many Ciphers have been developed to provide data security . The efficiency of the Ciphers that are being used depends mainly on their throughput and memory requirement. Using of large key spaces with huge number of rounds with multiple complex operations may provide security but at the same time affects speed of operation. Hence in this paper we have proposed a method to improve Caesar cipher with random number generation technique for key generation operations. The Caesar cipher has been expanded so as to include alphabets, numbers and symbols. The original Caesar cipher was restricted only for alphabets. The key used for Caesar Substitution has been derived using a key Matrix Trace value restricted to Modulo 94. The Matrix elements are generated using recursive random number generation equation, the output of which solely depends on the value of seed selected . In this paper, we made an effort to incorporate modern cipher properties to classical cipher. The second stage of encryption has been performed using columnar transposition with arbitrary random order column selection. Thus the proposed Scheme is a hybrid version of classical and modern cipher properties. The proposed method provides appreciable Security with high throughput and occupies minimum memory space. The Method is resistant against brute-force attack with 93! Combinations of keys, for Caesar encryption.
Improved Caesar Cipher with Random Number Generation Technique and Multistage...ijcisjournal
Secured Communication involves Encryption process at the sending end and Decryption process at the receiving end of the communication system. Many Ciphers have been developed to provide data security . The efficiency of the Ciphers that are being used depends mainly on their throughput and memory requirement. Using of large key spaces with huge number of rounds with multiple complex operations may provide security but at the same time affects speed of operation. Hence in this paper we have proposed a method to improve Caesar cipher with random number generation technique for key generation operations. The Caesar cipher has been expanded so as to include alphabets, numbers and symbols. The original Caesar cipher was restricted only for alphabets. The key used for Caesar Substitution has been derived using a key Matrix Trace value restricted to Modulo 94. The Matrix elements are generated using recursive random number generation equation, the output of which solely depends on the value of seed selected . In this paper, we made an effort to incorporate modern cipher properties to classical cipher. The second stage of encryption has been performed using columnar transposition with arbitrary random order column selection. Thus the proposed Scheme is a hybrid version of classical and modern cipher properties. The proposed method provides appreciable Security with high throughput and occupies minimum memory space. The Method is resistant against brute-force attack with 93! Combinations of keys, for Caesar encryption.
An Efficient Approach for Enhancing the Security of Amazigh Text using Binary...Editor IJCATR
Now a day’s Cryptography is one of the broad areas for researchers. Due to its importance, several cryptography techniques
are adopted by many authors to secure the data, but still there is a scope to improve the previous approaches. The main of our research
is to develop a novel Approach for enhancing the security of Amazigh Text using binary tree. The plaintext considered is the
combination of Unicode characters. This paper contributes in the area of elliptic curve cryptography by encrypting data using matrix
approach and using the concept of tree traversal method for enhancing the security of the encrypted points. The security goals were
enhanced by making it difficult for attacker to predicate a pattern as well as speed of the encryption/decryption scheme. The results
show strength of the algorithm.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Digital Tools and AI for Teaching Learning and Research
Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm.
1. Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 14
Novel Methods of Generating Self-Invertible Matrix for Hill
Cipher Algorithm
Bibhudendra Acharya bibhudendra@gmail.com
Department of Electronics & Communication Engineering
National Institute of Technology Rourkela
Rourkela, 769 008, India
Girija Sankar Rath gsrath@nitrkl.ac.in
Professor, Department of Electronics & Communication Engineering
National Institute of Technology Rourkela
Rourkela, 769 008, India
Sarat Kumar Patra skpatra@nitrkl.ac.in
Professor, Department of Electronics & Communication Engineering
National Institute of Technology Rourkela
Rourkela, 769 008, India
Saroj Kumar Panigrahy skp.nitrkl@gmail.com
Department of Computer Science & Engineering
National Institute of Technology Rourkela
Rourkela, 769 008, India
Abstract
In this paper, methods of generating self-invertible matrix for Hill Cipher algorithm
have been proposed. The inverse of the matrix used for encrypting the plaintext
does not always exist. So, if the matrix is not invertible, the encrypted text cannot
be decrypted. In the self-invertible matrix generation method, the matrix used for
the encryption is itself self-invertible. So, at the time of decryption, we need not to
find inverse of the matrix. Moreover, this method eliminates the computational
complexity involved in finding inverse of the matrix while decryption.
Keywords: Hill Cipher, Encryption, Decryption, Self-invertible matrix.
1. INTRODUCTION
Today, in the information age, the need to protect communications from prying eyes is greater
than ever before. Cryptography, the science of encryption, plays a central role in mobile phone
communications, pay-TV, e-commerce, sending private emails, transmitting financial information,
security of ATM cards, computer passwords, electronic commerce and touches on many aspects
of our daily lives [1]. Cryptography is the art or science encompassing the principles and methods
of transforming an intelligible message (plaintext) into one that is unintelligible (ciphertext) and
then retransforming that message back to its original form. In modern times, cryptography is
considered to be a branch of both mathematics and computer science, and is affiliated closely
with information theory, computer security, and engineering [2].
2. Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 15
Conventional Encryption is referred to as symmetric encryption or single key encryption. It can be
further divided into categories of classical techniques and modern techniques. The hallmark of
conventional encryption is that the cipher or key to the algorithm is shared, i.e., known by the
parties involved in the secured communication. Substitution cipher is one of the basic
components of classical ciphers. A substitution cipher is a method of encryption by which units of
plaintext are substituted with ciphertext according to a regular system; the units may be single
letters (the most common), pairs of letters, triplets of letters, mixtures of the above, and so forth.
The receiver deciphers the text by performing an inverse substitution [3]. The units of the plaintext
are retained in the same sequence as in the ciphertext, but the units themselves are altered.
There are a number of different types of substitution cipher. If the cipher operates on single
letters, it is termed a simple substitution cipher; a cipher that operates on larger groups of letters
is termed polygraphic. A monoalphabetic cipher uses fixed substitution over the entire message,
whereas a polyalphabetic cipher uses a number of substitutions at different times in the
message— such as with homophones, where a unit from the plaintext is mapped to one of
several possibilities in the ciphertext. Hill cipher is a type of monoalphabetic polygraphic
substitution cipher.
In this paper, we proposed novel methods of generating self-invertible matrix which can be used
in Hill cipher algorithm. The objective of this paper is to overcome the drawback of using a
random key matrix in Hill cipher algorithm for encryption, where we may not be able to decrypt
the encrypted message, if the matrix is not invertible. Also the computational complexity can be
reduced by avoiding the process of finding inverse of the matrix at the time of decryption, as we
use self-invertible key matrix for encryption.
The organization of the paper is as follows. Following the introduction, the basic concept of Hill
Cipher is outlined in section 2. Section 3 discusses about the modular arithmetic. In section 4,
proposed methods for generating self-invertible matrices are presented. Finally, section 5
describes the concluding remarks.
2. HILL CIPHER
It is developed by the mathematician Lester Hill in 1929. The core of Hill cipher is matrix
manipulations. For encryption, algorithm takes m successive plaintext letters and instead of that
substitutes m cipher letters. In Hill cipher, each character is assigned a numerical value like
25,...,1,0 === zba [4]. The substitution of ciphertext letters in the place of plaintext letters leads
to m linear equation. For 3=m , the system can be described as follows:
26mod)(
26mod)(
26mod)(
3332321311
3232221212
3132121111
PKPKPKC
PKPKPKC
PKPKPKC
++=
++=
++=
… (1)
This case can be expressed in terms of column vectors and matrices:
=
3
2
1
333231
232221
131211
3
2
1
P
P
P
KKK
KKK
KKK
C
C
C
… (2)
or simply we can write as KPC = , where C and P are column vectors of length 3, representing
the plaintext and ciphertext respectively, and K is a 33× matrix, which is the encryption key. All
operations are performed 26mod here. Decryption requires using the inverse of the matrix K .
The inverse matrix 1−
K of a matrix K is defined by the equation IKKKK --
== 11
, where I is
the Identity matrix. But the inverse of the matrix does not always exist, and when it does, it
satisfies the preceding equation. 1−
K is applied to the ciphertext, and then the plaintext is
recovered. In general term we can write as follows:
For encryption: pk KPEC == )( … (3)
3. Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 16
For decryption: PKKCKCDP p
--
k ==== 11
)( … (4)
3. MODULAR ARITHMETIC
The arithmetic operation presented here are addition, subtraction, unary operation, multiplication
and division [5]. Based on this, the self-invertible matrix for Hill cipher algorithm is generated. The
congruence modulo operator has the following properties:
1. ( )banpba −≡ ifmod
2. ( ) ( ) pbapbpa modmodmod ≡⇒=
3. pabpba modmod ≡⇒≡
4. pcapabpba modmodandmod ≡⇒≡≡
Let [ ]app −= ,...,1,0Z the set of residues modulo p . If modular arithmetic is performed within this
set pZ , the following equations present the arithmetic operations:
1. Addition : ( ) ( ) ( )[ ] ppbpapba modmodmodmod +=+
2. Negation : )mod(mod pappa −=−
3. Subtraction : ( ) ( ) ( )[ ] ppbpapba modmodmodmod −=−
4. Multiplication : ( ) ( ) ( )[ ] ppbpapba modmodmodmod ∗=∗
5. Division : ( ) cpba =mod/ when ( ) pcba mod∗=
The following Table exhibits the properties of modular arithmetic.
Property Expression
Commutative Law
( ) ( )
( ) ( ) pxpx
pxpx
modmod
modmod
ωω
ωω
∗=∗
+=+
Associative law ( )[ ] ( )[ ] pyxpyx modmod ++=++ ωω
Distribution Law ( )[ ] ( ){ } ( ){ }[ ] ppypxpyx modmodmodmod ∗∗∗=+∗ ωωω
Identities
( )
( ) papaand
papa
modmod1
modmod0
=∗
=+
Inverses
For each ,pZx ∈ ∃ y such that ( ) xypyx −==+ then0mod
For each pZx ∈ ∃ y such that ( ) 1mod =∗ pyx
Table 1: Properties of Modular Arithmetic
4. PROPOSED METHODS FOR GENERATING SELF-INVERTIBLE MATRIX
As Hill cipher decryption requires inverse of the matrix, so while decryption one problem arises
that is, inverse of the matrix does not always exist [5]. If the matrix is not invertible, then
encrypted text cannot be decrypted. In order to overcome this problem, we suggest the use of
self-invertible matrix generation method while encryption in the Hill Cipher. In the self-invertible
matrix generation method, the matrix used for the encryption is itself self-invertible. So, at the
time of decryption, we need not to find inverse of the matrix. Moreover, this method eliminates the
computational complexity involved in finding inverse of the matrix while decryption.
A is called self-invertible matrix if 1−
= AA . The analyses presented here for generation of self-
invertible matrix are valid for matrix of +ve integers, that are the residues of modulo arithmetic on
a prime number.
4. Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 17
4.1 Generation of self-invertible 22× matrix
Let
=
2221
1211
aa
aa
A , then,
)(tdeterminan
))((cofactor
)(tdeterminan
)(adjoint1
A
A
A
A
A
T
==−
−
−
∆
=∴ −
1121
12221 1
aa
aa
a
A , where, a∆ is the )(tdeterminan A
A is said to be self-invertible if 1−
= AA
So, aaa ∆−= /1212 & aaa ∆−= /2121
0and1 22112211 =+⇒−=−=∆∴ aaaaa … (5)
Example: (For modulo 13)
=
1112
32
A
4.2 Generation of self-invertible 33× matrix
Let
=
=
2221
1211
333231
232221
131211
AA
AA
aaa
aaa
aaa
A
where 11A is a 11× matrix = [ ]11a , 12A is a 21× matrix = [ ]1312 aa ,
21A is a 12× matrix =
31
21
a
a
and 22A is 22× matrix =
3332
2322
aa
aa
If A is self-invertible then,
IAAAAAAA
AAAAIAAA
=+=+
=+=+
2
22122121221121
221212112112
2
11
and,0
,0,
… (6)
Since 11A is 11× matrix = [ ]11a and ( ) 0221121 =+ AIaA
For non- trivial solution, it is necessary that 02211 =+ AIa
That is −=11a (one of the Eigen values of 22A )
1221AA can also be written as
=
=
13311231
132112211312
31
21
1221
000
0
aaaa
aaaaaa
a
a
AA
So 1221 AA is singular and
2
221221 AIAA −= … (7)
Hence 22A must have an Eigen value 1± . It can be shown that [ ] 21121221 AAAATrace = .
Since it can be proved that if −== 1111 aA (one of the Eigen values of 22A ),
then, any non-trivial solution of the equation (7) will also satisfy
2
112112 1 aAA −= … (8)
Example: (For modulo 13)
Take
=
61
52
22A which has Eigen value =λ 1 and 7
−=11a 7 = 6 or −1 = 12
If 611 =a ,
then,
=
−=
−=−=
125
125
28
19
61
52
61
522
221221 IIAIAA
5. Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 18
51221 =aa . So, 521 =a and 112 =a
121321 =aa . So, 5
5
12
13 ==a and 5
1
5
31 ==a
So the matrix will be
=
615
525
516
A . Other matrix can also be obtained if we take 1211 =a .
4.3 Generation of self-invertible 44× matrix
Let
=
44434241
34333231
24232221
14131211
aaaa
aaaa
aaaa
aaaa
A be self-invertible matrix partitioned as
=
2221
1211
AA
AA
A ,
where
=
2221
1211
11
aa
aa
A ,
=
2423
1413
12
aa
aa
A ,
=
4241
3231
21
aa
aa
A ,
=
4243
3433
22
aa
aa
A
Then, 2
112112 AIAA −= , 022121211 =+ AAAA ,
021221121 =+ AAAA , and 2
221221 AIAA −=
In order to obtain solution for all the four matrix equations, 2112 AA can be factorized as
( )( )11112112 AIAIAA +−= … (9)
So, if ( )kAIA 1112 −= or ( )kAI 11+
( )
k
AIA
1
1121 += or ( )
k
AI
1
11− , where k is a scalar constant.
Then, ( ) ( ) 2211111122121211 AkAIkAIAAAAA −+−=+ or ( ) ( )112211 AIAAk −+
So, IAorAA ==+ 112211 0 … (10)
Since IA =11 is a trivial solution, then, 02211 =+ AA is taken.
When we solve the 3
rd
and 4
th
matrix equations, same solution is obtained.
Example: (For Modulo 13)
Take
=
48
31
22A then,
=
95
1012
11A
Take 1112 AIA −= with 1=k . Then,
=
58
32
12A and
=
105
100
21A
So
=
48105
31100
5895
321012
A
4.4 A general method of generating an even self-invertible matrix
Let A=
nnnn
n
n
aaa
aaa
aaa
......
...............
...............
......
......
21
22221
11211
be an nn × self-invertible matrix partitioned to
=
2221
1211
AA
AA
A ,
where n is even and 2221,1211 &, AAAA are matrices of order
22
nn
× each.
6. Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 19
So, ( )( )1111
2
112112 AIAIAIAA +−=−= … (11)
If 12A is one of the factors of 2
11AI − then 21A is the other.
Solving the 2
nd
matrix equation results 02211 =+ AA .
Then form the matrix.
Algorithm:
1. Select any arbitrary
22
nn
× matrix 22A .
2. Obtain 2211 AA −=
3. Take ( ) ( )111112 or AIkAIkA +−= for k a scalar constant.
4. Then ( ) ( )111121
1
or
1
AI
k
AI
k
A −+=
5. Form the matrix completely.
Example: (For modulo 13)
Let
=
43
210
22A , then,
=
910
113
11A
If k is selected as 2,
=−=
106
49
)( 1112 AIkA and
=
55
122
21A
So,
=
4355
210122
106910
49113
A
4.5 A general method of generating self-invertible matrix
Let A=
nnnn
n
n
aaa
aaa
aaa
...
............
...
...
21
22221
11211
be an nn × self-invertible matrix partitioned to
=
2221
1211
AA
AA
A
11A is a 11× matrix = [ ]11a , 12A is a )1(1 −× n matrix = [ ]naaa 11312 ...
21A is a 1)1( ×−n matrix =
1
31
21
...
na
a
a
, 22A is a )1()1( −×− nn matrix =
nnnn
n
n
aaa
aaa
aaa
...
............
...
...
32
33332
22322
So, 2
11
2
112112 1 aAIAA −=−= … (12)
and ( ) 0221112 =+ AIaA … (13)
Also, −=11a (one of the Eigen values of 22A other than 1)
Since 1221AA is a singular matrix having the rank 1
and 2
221221 AIAA −= … (14)
So, 2
22A must have rank of )2( −n with Eigen values +1 of )2( −n multiplicity.
Therefore, 22A must have Eigen values 1± .
It can also be proved that the consistent solution obtained for elements 21A & 12A by solving the
equation (14) term by term will also satisfy the equation (12).
7. Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 20
Algorithm:
1. Select 22A , a non-singular )1()1( −×− nn matrix which has )2( −n number of Eigen values
of either +1 or −1 or both.
2. Determine the other Eigen value λ of 22A .
3. Set λ−=11a .
4. Obtain the consistent solution of all elements of 21A & 12A by using the equation (14).
5. Formulate the matrix.
Example: (For modulo 13)
Let
=
435
21012
1069
22A which has Eigen values 10,1±=λ
So, [ ]311 =A , and one of the consistent solutions of [ ]491112 =A and
=
5
2
10
21A
So,
=
4355
210122
106910
49113
A
Another consistent solution of [ ]112112 =A and
=
3
9
6
21A
So,
=
4353
210129
10696
11213
A
4.6 Another method to generate self-invertible matrix
Let A be any non-singular matrix and E be its Eigen matrix. Then we know that λEAE = ,
where λ is diagonal matrix with the Eigen values as diagonal elements. E the Eigen matrix is
non-singular.
Then, 1−
= EEA λ … (15)
and ( ) 1111111 −−−−−−−
=== EEEEEEA λλλ … (16)
So, 1−
= AA only when 1−
= λλ
If
=
nλ
λ
λ
λ
..0000
.......
.......
0..000
0..000
2
1
then,
=−
nλ
λ
λ
λ
1
000000
.......
.......
0..00
1
0
0...00
1
2
1
1
Thus 1−
= λλ when 1or
1
±== i
i
i λ
λ
λ
8. Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 21
Algorithm:
1. Select any nonsingular matrix E .
2. Form a diagonal matrix λ with 1±=λ but all value of λ must not be equal.
3. Then compute AEE =−1
λ .
Example: (For modulo 13)
=
−−
−−
=
= −
12212
2012
1276
121
2
3
13
3
10
1
3
8
3
8
,
887
654
321
1
EE
Take
=
100
0120
001
λ
=
== −
803
6110
505
12212
2012
1276
100
0120
001
857
684
3111
1
EEA λ
5. CONCLUSION
This paper suggests efficient methods for generating self-invertible matrix for Hill Cipher
algorithm. These methods encompass less computational complexity as inverse of the matrix is
not required while decrypting in Hill Cipher. These proposed methods for generating self-
invertible matrix can also be used in other algorithms where matrix inversion is required.
6. REFERENCES
1. Blakley G.R., “Twenty years of cryptography in the open literature”, Security and Privacy
1999, Proceedings of the IEEE Symposium, 9-12 May 1999
2. Imai H., Hanaoka G., Shikata J., Otsuka A., Nascimento A.C., “Cyptography with Information
Theoretic Security”, Information Theory Workshop, 2002, Proceedings of the IEEE, 20-25 Oct
2002
3. A. J. Menezes, P.C. Van Oorschot, S.A. Van Stone, “Handbook of Applied Cryptography”,
CRC press, 1996
4. W. Stallings, “Cryptography and Network Security”, 4
th
edition, Prentice Hall, 2005
5. Bruce Schneir, “Applied Cryptography”, 2
nd
edition, John Wiley & Sons, 1996
![Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 14
Novel Methods of Generating Self-Invertible Matrix for Hill
Cipher Algorithm
Bibhudendra Acharya bibhudendra@gmail.com
Department of Electronics & Communication Engineering
National Institute of Technology Rourkela
Rourkela, 769 008, India
Girija Sankar Rath gsrath@nitrkl.ac.in
Professor, Department of Electronics & Communication Engineering
National Institute of Technology Rourkela
Rourkela, 769 008, India
Sarat Kumar Patra skpatra@nitrkl.ac.in
Professor, Department of Electronics & Communication Engineering
National Institute of Technology Rourkela
Rourkela, 769 008, India
Saroj Kumar Panigrahy skp.nitrkl@gmail.com
Department of Computer Science & Engineering
National Institute of Technology Rourkela
Rourkela, 769 008, India
Abstract
In this paper, methods of generating self-invertible matrix for Hill Cipher algorithm
have been proposed. The inverse of the matrix used for encrypting the plaintext
does not always exist. So, if the matrix is not invertible, the encrypted text cannot
be decrypted. In the self-invertible matrix generation method, the matrix used for
the encryption is itself self-invertible. So, at the time of decryption, we need not to
find inverse of the matrix. Moreover, this method eliminates the computational
complexity involved in finding inverse of the matrix while decryption.
Keywords: Hill Cipher, Encryption, Decryption, Self-invertible matrix.
1. INTRODUCTION
Today, in the information age, the need to protect communications from prying eyes is greater
than ever before. Cryptography, the science of encryption, plays a central role in mobile phone
communications, pay-TV, e-commerce, sending private emails, transmitting financial information,
security of ATM cards, computer passwords, electronic commerce and touches on many aspects
of our daily lives [1]. Cryptography is the art or science encompassing the principles and methods
of transforming an intelligible message (plaintext) into one that is unintelligible (ciphertext) and
then retransforming that message back to its original form. In modern times, cryptography is
considered to be a branch of both mathematics and computer science, and is affiliated closely
with information theory, computer security, and engineering [2].](https://image.slidesharecdn.com/ijs2-160217194213/85/Novel-Methods-of-Generating-Self-Invertible-Matrix-for-Hill-Cipher-Algorithm-1-320.jpg)
![Bibhudendra Acharya, Girija Sankar Rath, Sarat Kumar Patra, and Saroj Kumar Panigrahy
International Journal of Security, Volume 1 : Issue (1) 15
Conventional Encryption is referred to as symmetric encryption or single key encryption. It can be
further divided into categories of classical techniques and modern techniques. The hallmark of
conventional encryption is that the cipher or key to the algorithm is shared, i.e., known by the
parties involved in the secured communication. Substitution cipher is one of the basic
components of classical ciphers. A substitution cipher is a method of encryption by which units of
plaintext are substituted with ciphertext according to a regular system; the units may be single
letters (the most common), pairs of letters, triplets of letters, mixtures of the above, and so forth.
The receiver deciphers the text by performing an inverse substitution [3]. The units of the plaintext
are retained in the same sequence as in the ciphertext, but the units themselves are altered.
There are a number of different types of substitution cipher. If the cipher operates on single
letters, it is termed a simple substitution cipher; a cipher that operates on larger groups of letters
is termed polygraphic. A monoalphabetic cipher uses fixed substitution over the entire message,
whereas a polyalphabetic cipher uses a number of substitutions at different times in the
message— such as with homophones, where a unit from the plaintext is mapped to one of
several possibilities in the ciphertext. Hill cipher is a type of monoalphabetic polygraphic
substitution cipher.
In this paper, we proposed novel methods of generating self-invertible matrix which can be used
in Hill cipher algorithm. The objective of this paper is to overcome the drawback of using a
random key matrix in Hill cipher algorithm for encryption, where we may not be able to decrypt
the encrypted message, if the matrix is not invertible. Also the computational complexity can be
reduced by avoiding the process of finding inverse of the matrix at the time of decryption, as we
use self-invertible key matrix for encryption.
The organization of the paper is as follows. Following the introduction, the basic concept of Hill
Cipher is outlined in section 2. Section 3 discusses about the modular arithmetic. In section 4,
proposed methods for generating self-invertible matrices are presented. Finally, section 5
describes the concluding remarks.
2. HILL CIPHER
It is developed by the mathematician Lester Hill in 1929. The core of Hill cipher is matrix
manipulations. For encryption, algorithm takes m successive plaintext letters and instead of that
substitutes m cipher letters. In Hill cipher, each character is assigned a numerical value like
25,...,1,0 === zba [4]. The substitution of ciphertext letters in the place of plaintext letters leads
to m linear equation. For 3=m , the system can be described as follows:
26mod)(
26mod)(
26mod)(
3332321311
3232221212
3132121111
PKPKPKC
PKPKPKC
PKPKPKC
++=
++=
++=
… (1)
This case can be expressed in terms of column vectors and matrices:
=
3
2
1
333231
232221
131211
3
2
1
P
P
P
KKK
KKK
KKK
C
C
C
… (2)
or simply we can write as KPC = , where C and P are column vectors of length 3, representing
the plaintext and ciphertext respectively, and K is a 33× matrix, which is the encryption key. All
operations are performed 26mod here. Decryption requires using the inverse of the matrix K .
The inverse matrix 1−
K of a matrix K is defined by the equation IKKKK --
== 11
, where I is
the Identity matrix. But the inverse of the matrix does not always exist, and when it does, it
satisfies the preceding equation. 1−
K is applied to the ciphertext, and then the plaintext is
recovered. In general term we can write as follows:
For encryption: pk KPEC == )( … (3)](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)